1. René Descartes was a brilliant man, and helped to advance mathematics in many ways. He helped advance the area of geometry the most, and stated that any problem could be expressed using one or more axes, curves, and lines. He also began using a and b to express points to the left or the right of a given axis. He helped to collapse the barriers between math and science, creating analytical geometry.He was a pioneer for "universal mathematics", a movement that melded all mathematics together, rather than seperate them into various, small categories.
He was also the first person to begin using exponents to represent a number times itself n times.He also created Descartes' Rule of Signs, which made it easier to find how many real roots a polynomial posses, based on the sign in front of a given element of the polynomial.
2. I believe that Descartes' pioneering for universal mathematics was the most important thing he did, because it opened up mathematics for much more improvement and application. By bringing all the smaller pieces together, he made mathematics much more approachable and increased overall general knowledge of math.
3. Maria Gaetana Agnesi (1718-1799) was the oldest of 21 children that her father had with three wives. I find that particularly interesting. (found on http://www.agnesscott.edu/lriddle/women/agnesi.htm)
4. While it is not necessarily a new topic for me, I do find the Midpoint Formula to be the most useful of the formulas in section 8.1. (image found on www.cliffsnotes.com)
When analyzing graphs (such as a graph on your business growth, profits, and so forth) it would be very beneficial to know the exact midpoint. With this equation, the exact midpoint will always be found without doubt.
5. m= -1/2, line passes through (-2, 4).
I started from point (-2, 4). Using the slope, I went down 1 and 2 to the right. I did this 3 or so times. I already knew the slope, so I just needed to look for the y intercept.
The final equation is y=(-1/2)x+3
April 29, 2011
1. I believe that Johannes Kepler was more of a scientist than a mathematician, because he was always seeking ways to apply his intelligence to scientific fields, such as astronomy. Kepler used conic sections to describe the things around us, like our solar system. Kepler's first law of planetary motion, for instance, uses ellipticals to describe the path that planets take around the sun.
However, this is not to say that he did not contribute to mathematics. He gave the first proof as to how logarithms work, using Rudolphine Tables. He also created the first mathematical theory as to how a camera obscura (telescope) might work, and was aslo the first to correctly explain the workings of a human eye. These contributions still have an impact on many aspects of the world today.
2. Logarithms are used in many real-world applications. For instance, the Richter Scale (the scale to measure the intensity of an earthquake) uses a logarithm. Also, the scale used to measure loudness of a sound (in decibles) also uses a logarithm. This shows how logarithms are very useful in explaining the physical world around us. Logarithms can also be used to help you understand monetary investments, for they are also used in calculating interest rates.
3. When we began 8.7, I had many problems solving a system of equations by substitution. However, as we progressed further into the section and the chapter, I soon learned how to solve a system of equations using substitution.
Example- Solve the system:
3x+2y=13 (1)
4x-y=-1 (2)
Solve for y
-y=-1-4x move y to one side of the equation by itelf y= 1+4x Now, substitute 1+4x into any place a y is in equation 1 3x+2(1+4x)=13 3x+2+8x=13
11x=11 x=1 Now I will plug 1 into any place an x is in equation 2 4(1)-y=-1 -y=-5 y=5
To check my answer, I plug 1 into every x place and 5 into every y place in either equation.
3(1)+2(5)=13
3+10=13 13=13
Therefore, the final answer is (1,5)
4. To satisfy the solution of (3,-4), I created these two equations:
2x+y=2 x+2y=-5
To solve this equation, I first use the elimination method. I multiply the first equation by -2:
(-2) 2x+y=2
-4x-2y=-4
Then, I align the second equation below the newly modified equation and add as if an addition problem:
-4x-2y=-4
+ x+2y=-5
-3x=-9
Therefore, x=3 Now I plug 3 into every x place in the second equation
3+2y=-5
Subtract 3 from each side:
2y=-8
Therefore, y=-4
I check my equation by plugging in 3 into every x place and -4 into every y place in either of the original two equations:
2(3)+(-4)=2
6-4=2 2=2 Finally, it is safe to say that the two values are (3,-4)
5. If I were to try and explain how to solve a system of linear inequalities to a student who does not know how, I would first make sure they knew that it was nothing to be stress about.
Just solve for y:
3x+2y>6
Subtract 3x from both sides
2y>-3x+6
Now, get y by itself by dividing both sides by 2 y>-3/2x+6 Now, you may notice that this looks a lot like slope-intercept form. In fact, it can be graphed just as if it were in slope-intercept form. However, in linear inequalities, there are two extra things to pay attention to:
1) Whether or not the line is solid or dotted. Use a solid line in cases of greater/lesser than or equal to, and use a dotted line if it is just greater/lesser than.
2) Shading. Find a point that does not lie on the line, such as (0,0). Now, plug in its x and y values in the orignal inequality.
3(0)+2(0)>6
0>6 is false, so shade on the side of the line in which the point (0,0) does not lie.
Now, to make it a system of linear inequalities, just add one more linear inequality, such as:
2x-5y<10
Get y by itself:
-5y<-2x+10
Now, since you'll be dividing by a negative, you will switch the inequality sign around: y>2/5x-2
Now, graph this line in the exact method as before, using the same shading method. You will notice that on the graph, there will be a space where both shadings from the lines meet. This is your solution set.
May 13, 2011
1. To the Greeks, geometry was the pinnacle of mathematics. The word geometry, in fact, comes from the Greek words Geo (earth) and Metria (measurement). Geometry for the Greeks was very intuitive, meaning that the mathematicians looked for facts relating to the angle or shape without planning to demonstrate or prove them using deductive reasoning. Many Greek schools of mathematics, including the Platonic school, the Sophist school, and the Pythagorean School taught their students to use mathematics as a tool to understand the universe, and that it has almost infinite applications.When the Greeks made contructions, they followed a strict code of conduct.They were not to line up points based on the eye, and they were to only use two tools. These tools were the straight-edge (without marks) to draw line segments and the compass to draw arcs and circles. They used these two tools, also to attempt to solve the Three Problems of Antiquity, which are:1) To square a circle. Given the area of a circle, construct a square equal in area to the circle.2) To double a cube. Given the length of one side of a cube, construct a second cube double in volume of the first.3) To trisect an angle. Using an arbitrary angle, divide it into three equal angles.All three of these problems, more than 2000 years after they were propsed, were proved unsolvable using only a straight-edge and a compass.2. Shapes in geometry are often used in the real world based on their properties. For instance, a cirlce is used as a wheel because of its even radii and smooth surface. Squares and rectangles are often used in architecture for their structural stability. Triangles are used in constructing many bridges, because they provide a very strong, but rigid, structure. Cylinders are often used because they hold an optimum volume of product and use a minimal amount of space.3. Euclid was a Greek mathematician who lived around 300 B.C. Little is known about his life. However, it is known that hye did much to advance mathematics, especially in the area of geometry.His books, called Elements, were a 13 book series that covered geometry and other areas of mathematics. In Elements, Euclid uses a deductive system to present his problems and solutions, which was revolutionary in the world of mathematics. If I could ask him one question, it would be "why geometry?". There are many fields of math, and I would like to know what drove him to contribute largely to the area of geometry.4. A parallelogram, by definition, is a quadrilateral with two pair of parallel sides. The sides opposite to one another, in order to be parallel, must also be equal. A rhombus, by definition, is a quadrilateral whose all four sides have equal length. Therefore, a rhombus must be a parallelogram, because the sides opposite to one another must be equal, therefore parallel. However, a parallelogram is not always a rhombus because it's sides opposite to one another must be congruent.
April 15, 2011
1. René Descartes was a brilliant man, and helped to advance mathematics in many ways. He helped advance the area of geometry the most, and stated that any problem could be expressed using one or more axes, curves, and lines. He also began using a and b to express points to the left or the right of a given axis. He helped to collapse the barriers between math and science, creating analytical geometry.He was a pioneer for "universal mathematics", a movement that melded all mathematics together, rather than seperate them into various, small categories.He was also the first person to begin using exponents to represent a number times itself n times.He also created Descartes' Rule of Signs, which made it easier to find how many real roots a polynomial posses, based on the sign in front of a given element of the polynomial.
2. I believe that Descartes' pioneering for universal mathematics was the most important thing he did, because it opened up mathematics for much more improvement and application. By bringing all the smaller pieces together, he made mathematics much more approachable and increased overall general knowledge of math.
3. Maria Gaetana Agnesi (1718-1799) was the oldest of 21 children that her father had with three wives. I find that particularly interesting. (found on http://www.agnesscott.edu/lriddle/women/agnesi.htm)
4. While it is not necessarily a new topic for me, I do find the Midpoint Formula to be the most useful of the formulas in section 8.1.
When analyzing graphs (such as a graph on your business growth, profits, and so forth) it would be very beneficial to know the exact midpoint. With this equation, the exact midpoint will always be found without doubt.
5. m= -1/2, line passes through (-2, 4).
I started from point (-2, 4). Using the slope, I went down 1 and 2 to the right. I did this 3 or so times. I already knew the slope, so I just needed to look for the y intercept.
The final equation is y=(-1/2)x+3
April 29, 2011
1. I believe that Johannes Kepler was more of a scientist than a mathematician, because he was always seeking ways to apply his intelligence to scientific fields, such as astronomy. Kepler used conic sections to describe the things around us, like our solar system. Kepler's first law of planetary motion, for instance, uses ellipticals to describe the path that planets take around the sun.However, this is not to say that he did not contribute to mathematics. He gave the first proof as to how logarithms work, using Rudolphine Tables. He also created the first mathematical theory as to how a camera obscura (telescope) might work, and was aslo the first to correctly explain the workings of a human eye. These contributions still have an impact on many aspects of the world today.
2. Logarithms are used in many real-world applications. For instance, the Richter Scale (the scale to measure the intensity of an earthquake) uses a logarithm. Also, the scale used to measure loudness of a sound (in decibles) also uses a logarithm. This shows how logarithms are very useful in explaining the physical world around us. Logarithms can also be used to help you understand monetary investments, for they are also used in calculating interest rates.
3. When we began 8.7, I had many problems solving a system of equations by substitution. However, as we progressed further into the section and the chapter, I soon learned how to solve a system of equations using substitution.
Example- Solve the system:
3x+2y=13 (1)
4x-y=-1 (2)
Solve for y
-y=-1-4x move y to one side of the equation by itelf
y= 1+4x
Now, substitute 1+4x into any place a y is in equation 1
3x+2(1+4x)=13
3x+2+8x=13
11x=11
x=1
Now I will plug 1 into any place an x is in equation 2
4(1)-y=-1
-y=-5
y=5
To check my answer, I plug 1 into every x place and 5 into every y place in either equation.
3(1)+2(5)=13
3+10=13
13=13
Therefore, the final answer is (1,5)
4. To satisfy the solution of (3,-4), I created these two equations:
2x+y=2
x+2y=-5
To solve this equation, I first use the elimination method. I multiply the first equation by -2:
(-2) 2x+y=2
-4x-2y=-4
Then, I align the second equation below the newly modified equation and add as if an addition problem:
-4x-2y=-4
+
x+2y=-5
-3x=-9
Therefore, x=3
Now I plug 3 into every x place in the second equation
3+2y=-5
Subtract 3 from each side:
2y=-8
Therefore, y=-4
I check my equation by plugging in 3 into every x place and -4 into every y place in either of the original two equations:
2(3)+(-4)=2
6-4=2
2=2
Finally, it is safe to say that the two values are (3,-4)
5. If I were to try and explain how to solve a system of linear inequalities to a student who does not know how, I would first make sure they knew that it was nothing to be stress about.
Just solve for y:
3x+2y>6
Subtract 3x from both sides
2y>-3x+6
Now, get y by itself by dividing both sides by 2
y>-3/2x+6
Now, you may notice that this looks a lot like slope-intercept form. In fact, it can be graphed just as if it were in slope-intercept form. However, in linear inequalities, there are two extra things to pay attention to:
1) Whether or not the line is solid or dotted. Use a solid line in cases of greater/lesser than or equal to, and use a dotted line if it is just greater/lesser than.
2) Shading. Find a point that does not lie on the line, such as (0,0). Now, plug in its x and y values in the orignal inequality.
3(0)+2(0)>6
0>6 is false, so shade on the side of the line in which the point (0,0) does not lie.
Now, to make it a system of linear inequalities, just add one more linear inequality, such as:
2x-5y<10
Get y by itself:
-5y<-2x+10
Now, since you'll be dividing by a negative, you will switch the inequality sign around:
y>2/5x-2
Now, graph this line in the exact method as before, using the same shading method. You will notice that on the graph, there will be a space where both shadings from the lines meet. This is your solution set.
May 13, 2011
1. To the Greeks, geometry was the pinnacle of mathematics. The word geometry, in fact, comes from the Greek words Geo (earth) and Metria (measurement). Geometry for the Greeks was very intuitive, meaning that the mathematicians looked for facts relating to the angle or shape without planning to demonstrate or prove them using deductive reasoning. Many Greek schools of mathematics, including the Platonic school, the Sophist school, and the Pythagorean School taught their students to use mathematics as a tool to understand the universe, and that it has almost infinite applications.When the Greeks made contructions, they followed a strict code of conduct.They were not to line up points based on the eye, and they were to only use two tools. These tools were the straight-edge (without marks) to draw line segments and the compass to draw arcs and circles. They used these two tools, also to attempt to solve the Three Problems of Antiquity, which are:1) To square a circle. Given the area of a circle, construct a square equal in area to the circle.2) To double a cube. Given the length of one side of a cube, construct a second cube double in volume of the first.3) To trisect an angle. Using an arbitrary angle, divide it into three equal angles.All three of these problems, more than 2000 years after they were propsed, were proved unsolvable using only a straight-edge and a compass.2. Shapes in geometry are often used in the real world based on their properties. For instance, a cirlce is used as a wheel because of its even radii and smooth surface. Squares and rectangles are often used in architecture for their structural stability. Triangles are used in constructing many bridges, because they provide a very strong, but rigid, structure. Cylinders are often used because they hold an optimum volume of product and use a minimal amount of space.3. Euclid was a Greek mathematician who lived around 300 B.C. Little is known about his life. However, it is known that hye did much to advance mathematics, especially in the area of geometry.His books, called Elements, were a 13 book series that covered geometry and other areas of mathematics. In Elements, Euclid uses a deductive system to present his problems and solutions, which was revolutionary in the world of mathematics. If I could ask him one question, it would be "why geometry?". There are many fields of math, and I would like to know what drove him to contribute largely to the area of geometry.4. A parallelogram, by definition, is a quadrilateral with two pair of parallel sides. The sides opposite to one another, in order to be parallel, must also be equal. A rhombus, by definition, is a quadrilateral whose all four sides have equal length. Therefore, a rhombus must be a parallelogram, because the sides opposite to one another must be equal, therefore parallel. However, a parallelogram is not always a rhombus because it's sides opposite to one another must be congruent.